From: Martin Davis
>This question presupposes that any kind of "existence" must be existence
>somewhere. ...
>If the real number e exists, and we can't locate it in the
>physical world, then where does it exist?
>>So I think it will be with the locale of the objects of mathematics. It
>will come to be understood that, just as light can undulate perfectly
>well without anything to undulate in, so the objects of mathematics can
>have a perfectly satisfactory existence without any particular place in
>which to exist.
I agree wholeheartedly with Martin here, but would add two points.
First, even material existence is no guarantee of locatability.
Asimov tells the story of a Univac computer sold by an enthusiastic
salesman to Buddhist monks to speed up their Tower of Hanoi enumeration,
which the monks believe will take the lifetime of the universe to
complete. He gets it running for the monks, and tells the story to
his seat mate on the plane home to the US. As the story ends, they are
looking out the window and the stars start blinking out.
This story presupposes that the universe exists. But where? You can't
locate the universe in the physical world, it doesn't have a location,
it's everywhere.
If taking the whole universe seems a fishy example, consider instead
the cosmic background radiation, which is certainly not everywhere---it
does not exist underground for example, and Penzias et al received the
Nobel prize for finding it. So where did they find it? It is no more
locatable than the whole universe, though it does have an orientation
as determined by Bracewell et al.
Second, pursuing Martin's point that waves have no location, I claim that
the mathematical universe mirrors the physical universe in more detail
than it is customarily given credit for. I would view the points of
a topological space as having locations within that space, but I would
not so view the open sets, which act globally as predicates on that space.
With regard to locatability within a topological space, its points
are particle-like while its open sets are wave-like.
Where does the space itself exist? Unless you have a space of spaces, the
question is incoherent, location being an intrinsically spatial notion.
But our instinct to locate all ostensibly substantive entities, even
spaces themselves, leads us to postulate just such spaces of spaces.
Such a space is the category of topological spaces, whose homsets (not
individual morphisms) measure distance: Lawvere in a celebrated 1974
paper interpreting enriched categories geometrically pointed out that
every category forms a metric space satisfying a triangle inequality in
a suitable general sense.
Returning to waves, while classical predicates can only be square waves,
it adds diversity and interest to logic to permit predicates like "hot"
to vary continuously in space. Here "continuous" can be defined with
respect to a suitable topology, either a concrete topology such as that
of the reals as proposed by Zadeh, or better, a more abstract topology as
seemingly conceived by the self-professed mystic Brouwer. Continuously
variable truth is sufficiently hard to formulate set-theoretically that
those wedded to set theoretic foundations tend to reject continuous
predicates as incoherent, and I would agree that simply valuing them
in the real interval [0,1] is too naive.
In striking contrast, category theory provides an ideal foundation for
continuity in truth. Set theory and category theory are not equivalent,
only dual, and continuity in truth is one of those things that are made
much easier by dualizing the foundations of mathematics. For this and
other reasons I would strongly recommend that physicists equally weight
set-theoretic and category-theoretic foundations for the mathematics of
their subject.
Going even further out on a limb (one I have been sitting precariously
on for several years), I would make this particle-wave claim for *all*
concrete structures when represented, as I have been advocating, as
Chu spaces. Chu spaces generalize topological spaces in a way that
is not unreasonable (just drop the restrictions on the open sets of
closure-under-union-and-intersection and two-valued membership of their
points), yet is nevertheless sufficient to represent *all* the concrete
objects of mathematics as we understand it, fully, faithfully, and
concretely in the sense of my previous message. This generalization
retains the above sense of open sets as waves, and unlike ordinary
topology does not insist that they be square waves.
I regard points of view such as these to be foundational in the sense that
they concern the basic premises of our mathematical thought processes.
Lots of people on this list seem not to view these viewpoints as
foundational, regarding the foundations of mathematics as too cast
in concrete by now to admit of any such radical upheaval. I would be
interested in hearing how people would *substantively* attack the claim
that these alternative viewpoints are serious foundational proposals.
So far I am aware of only two attacks on the foundational nature of
such viewpoints that have been made consistently on this list.
1. The view is incoherent.
2. The holder of the view is a charlatan who must be coerced to recant.
Attack 2 is absurd. Attack 1 may be reasonable the first few times but
becomes tedious after a while. I urge the contributors to this list to
move on to attacks with more substance.
Better yet would be an honest effort to actually appreciate the viewpoints
we have been so earnestly advocating, if only so that you can see where
these viewpoints come from. They make much more sense than you give
them credit for.
Vaughan Pratt